Wilcoxon signed-rank test

A rank-based test for paired data (or one sample against a hypothesised median) using the ranks of the absolute differences and their signs.

W^+ = \sum_{d_i > 0} \operatorname{rank}(|d_i|),\quad W^- = \sum_{d_i < 0} \operatorname{rank}(|d_i|),\quad T = \min(W^+, W^-)

T is the two-sided statistic; W⁺ is used for one-sided tests. Zero differences are dropped (default) before ranking.

$d_i$: Paired difference $x_i - y_i - \Delta_0$ (or $x_i - m_0$ for one sample).
$n$: Number of nonzero differences.

\mu_T = \frac{n(n+1)}{4},\qquad \sigma_T^2 = \frac{n(n+1)(2n+1) - \tfrac12\sum t(t^2-1)}{24}

Normal approximation (with ties or zeros); exact null distribution of W⁺ otherwise for n ≤ 50.

$t$: Size of each group of tied |d|.

When to use

Paired measurements (before/after, matched pairs) when the differences are clearly non-normal with a small sample, or the outcome is ordinal-ish but differences can be ranked.

Assumptions

Pairs independent of each other.
The distribution of differences is symmetric about its median (needed to interpret the test as being about the median difference). If not, use the sign test.

How to interpret

H₀: differences are symmetric about 0 (median difference 0). Effect size: matched-pairs rank-biserial $r = (W^+ - W^-)/(W^+ + W^-)$ , from −1 to 1.

Common mistakes

Using it on two independent groups (use Mann–Whitney).
Forgetting that zero differences are dropped, which reduces n.
Expecting an "exact" p-value when there are ties or zeros.

Example

Before 140, 135, 150, 142, 138, 145; after 136, 132, 143, 141, 133, 139. Differences 4, 3, 7, 1, 5, 6 are all positive: $W^+ = 21$ , $W^- = 0$ , $T = 0$ . Exact two-sided $p = 2/64 = 0.031$ ; $r = 1$ (every pair decreased).